{% extends 'homepage.html' %}

{% block content %}


{% if degree==1 %}
<div>There are only {{ KNOWL('lfunction.alldegree1.proof', title='two types') }} of degree 1 L-functions:</div>
<h3>1. The <a href="/L/Riemann">Riemann zeta function</a></h3>
<div>
The {{ KNOWL('lfunction.riemann', title='Riemann zeta function,') }}
<a href="/L/Riemann">$\zeta(s)$</a>, was introduced by {{ KNOWL('bio.riemann', title='Riemann') }} in the mid-1800s as a tool to study
the distribution of prime numbers.
</div>

<h3>2. Dirichlet L-functions associated to a primitive {{ KNOWL('character.dirichlet', title='Dirichlet character') }}</h3>
<div>
{{ KNOWL('lfunction.dirichlet', title='Dirichlet L-functions') }} were introduced by {{ KNOWL('bio.dirichlet', title='Dirichlet') }} in the mid-1800s as a tool to study
prime numbers in arithmetic progressions.
</div>
<div>
These L-functions have a functional equation of the form 
\[
\Lambda(s,\chi) = q^{s/2} \Gamma_{\mathbb R} (s+a) L(s,\chi) = \varepsilon_\chi \overline{\Lambda}(1-s),
\]
where $q$ is the conductor of $\chi$.
</div>
<div>
The table below shows the primitive characters arranged by the conductor and the order of the
character, with color indicating the sign of the functional equation.
</div>

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{% endif %}

{% if degree==2 %}

<h2>Automorphic Objects</h2>

<div>There are two known types of primitive degree 2 L-functions that arise from automorphic objects:</div>
<ol>
<li><a href="degree2#GL2_Q_Holomorphic">$L(s, f)$: the L-function of a holomorphic newform in $S_k(\Gamma_0(N))$</a>
</li>
<li><a href="degree2#GL2_Q_Maass">$L(s, f)$: the L-function of a Maass cusp form on  $\Gamma_0(N)$</a></li>
</ol>

<h2>Geometric Objects</h2>

<div>There is one known type of primitive degree 2 L-function that arises from a geometric object:</div>
<ul>
<li>
<a href="#EllipticCurve_Q">$L(s, E)$: the L-function of an elliptic curve $E$ over $\mathbb{Q}$.</a></li>
</ul>
<h2>Available Examples</h2>

<h3 id="GL2_Q_Holomorphic">L-functions of holomorphic cusp forms with trivial character</h3>

<div>
These L-functions have a functional equation of the form 
\begin{equation} 
\Lambda_f(s) := N^{s/2} \Gamma_{\mathbb{C} }
\left(s + \frac{k-1}{2} \right) L(s, f) 
\pm \Lambda_f(1-s)
\end{equation}
If \(L(s) = \sum a_n n^{-s} \) then \(a_n n^{\frac{k-1}{2} } \) 
is an algebraic integer. 
The pairs \((N,k)\) for such L-functions are shown in the plot below.
<p>
<i>The color indicates the sign of the functional equation.  
The horizontal grouping indicates the degree of the field containing 
the arithmetically normalized coefficients. </i>
See the {{ KNOWL('lfunction.degree2holo.key', title='legend') }} for more details.
</p>
</div>


{{contents[1]|safe}}

<h3 id="GL2_Q_Maass">Maass cusp forms on GL(2)</h3>

{{contents[2]|safe}}

<h3 id="EllipticCurve_Q">Elliptic Curves over $\mathbb{Q}$</h3>

{{contents[0]|safe}}



{% endif %}

{% if degree==3 %}
<div>The known types of primitive degree 3 L-functions are:</div>
<p>
<div>$L(s, f, \mathrm{sym}^2)$: the symmetric square of a degree 2 L-function</div>
 <p>
<div>$L(s,f)$: standard L-function of a cusp form on GL(3)</div>


<h2>Available Examples:</h2>

{{contents|safe}}

{% endif %}
 
{% if degree==4 %}
<div>The known types of primitive degree 4 L-functions are:</div>
<p>
<div>$L(s, f, \mathrm{sym}^3)$: the symmetric cube L-function of a cusp form on GL(2)</div>
        <p>

        <div>$L(s, f, \mathrm{sym}^3)$: the symmetric cube L-function of a cusp form on GL(2)</div>
        <p>
	<div>$L(s, F, \mathrm{spin})$: spin L-function of a Siegel modular form on GSp(4) (genus 2)</div>
	<p>
	<div>$L(s,f)$: standard L-function of a cusp form on the Picard group SL(2,Z[i])</div>
	<p>
	<div>$L(s,f)$: standard L-function of a cusp form on Sp(4)</div>

	<p>
	<div>$L(s,f)$: standard L-function of a cusp form on GL(4)</div>
	<p>
	<div>$L(s,E)$: L-function of an elliptic curve over a quadratic field</div>
	<p>
	<div>$L(s,f)$: standard L-function of a Hilbert modular form on GL(2) over a real quadratic field</div>
        <p>

<h2>Available Examples:</h2>

<!-- <h3>Maass cusp forms on Sp(4,Z)</h3>
<div>These satisfy a functional equation with $\Gamma$-factors
\begin{equation}
\Gamma_R(s + i \mu_1) 
\Gamma_R(s + i \mu_2) 
\Gamma_R(s - i \mu_1) 
\Gamma_R(s - i \mu_2) 
\end{equation}
with $0 \le \mu_2 \le \mu_1$.
</div>
<div>The dots in the plot correspond to $(\mu_1,\mu_2)$ for Sp(4,Z) L-functions
which have been found by a computer search.</div>
<embed src="/static/images/Sp4.svg" width="470" height="330">
<br>
<h3>Maass cusp forms on SL(4,Z)</h3>
<div>These satisfy a functional equation with $\Gamma$-factors
\begin{equation}
\Gamma_R(s + i \mu_1) 
\Gamma_R(s + i \mu_2) 
\Gamma_R(s - i \mu_3) 
\Gamma_R(s - i \mu_4) 
\end{equation}
where $\mu_1 + \mu_2 = \mu_3 + \mu_4$.
</div>
<div>The dots in the plot correspond to $(\mu_1,\mu_2)$ for SL(4,Z) L-functions
which have been found by a computer search.</div>
<embed src="/static/images/SL4.svg" width="520" height="360"> -->

{{contents|safe}}

{% endif %}
 
{% endblock %}


